[size=150]A {k,n} regular tessellation is defined as a tiling by regular n-gons (convex regular polygons that cover all the space) where k polygons of n sides meet at each vertex. k, n [math]\ge[/math] 3.[br]On the Euclidean Plane the only choices are {6,3}, {4,4} and {3,6} regular tessellations by traingles, squares or hexagons.[br][br][b]Theorem of Classification.[br][/b]For any k,n [math]\ge[/math] 3, there exists a tiling on the one type of geometry depending: [br][list][*] If [i] (1/k) + (1/n) > 1/2[/i], then the tilling is spherical[br][/*][*] If (1/k) + (1/n) = 1/2, then the tilling is euclidean[br][/*][*] If (1/k) + (1/n) < 1/2, then the tilling is hyperbolic.[/*][/list][b][i][br]Consequence.[/i][/b][br]If we dealing with triangles [i](n= 3), then when k = 3, 4 or 5 [/i]we get an spherical tiling. For[i] k = 6 [/i]we get an euclidean tiling, and for[i] k [math]\ge[/math] 7 [/i]we get an hyperbolic tessellation.[br][br]As an example we can have {7,3} and {8,3] tessellations. This last one in Circle Limit III.[br][br][/size]